[Math] Find the Bounding Rectangle of Rotated Rectangle

coordinate systemsgeometryrotations

I have rectangle with co-ordinates(x1,y1) and (x2,y2) and I have to rotate the rectangle an amount of θ about it centre using Rotation Matrix

 |  cosθ  sinθ |
 | -sinθ  cosθ |

I need to find the co-ordinates of bounding rectangle after rotation.

Before rotation

0,0
 |"""""""""""""""""""""""""""""""""""""""""""|
 |                                           |
 |  x1,y1                                    |
 |       |"""""""""""""|                     |
 |       |             |                     |
 |       |             |                     | 
 |       |             |                     |
 |       """""""""""""" x2,y2                |
 |                                           |
 |                                           |
  """"""""""""""""""""""""""""""""""""""""""" W,H

After rotation

 0,0
     |"""""""""""""""""""""""""""""""""""""""""""|
     |           ?,?                             |
     |            |""""/\"""""|                  |      
     |            |   /   \   |                  |
     |            |  /      \ |                  |
     |            | /        /|                  |
     |            |/        / |                  |
     |            |\       /  |                  |
     |            |  \    /   |                  |
     |            |    \ /    |                  |
     |             """""""""""  ?,?              |
     |                                           |
     |                                           |
      """"""""""""""""""""""""""""""""""""""""""" W,H

Is there any general equation for finding the co-ordinates of bounding rectangle?.

Thanks….

Haris.

Best Answer

For each of the four original corners: $$ (x_1,y_1), (x_1,y_2), (x_2,y_1), (x_2,y_2) $$ use the rotation matrix to obtain four new corners. Then to obtain the two ordered pairs that define your bounding rectangle, let one ordered pair have the minimum $x$ and $y$ values out of all the new corners and let the other ordered pair have the maximum $x$ and $y$ values.